Math Problem Statement

Let S be a set of n elements, then find out the number of ordered pairs in the largest and the smallest equivalence relations on S

Solution

The problem is asking about equivalence relations on a set SS with nn elements, specifically for the largest and the smallest equivalence relations. Let's break this down:

Equivalence Relation

An equivalence relation on a set is a relation that is:

  1. Reflexive (every element is related to itself),
  2. Symmetric (if aa is related to bb, then bb is related to aa),
  3. Transitive (if aa is related to bb and bb is related to cc, then aa is related to cc).

Number of Ordered Pairs in a Relation

In general, a relation on a set SS of nn elements can be thought of as a subset of the Cartesian product S×SS \times S, which consists of all ordered pairs (a,b)(a, b), where a,bSa, b \in S. The total number of ordered pairs in any relation is the size of the relation, i.e., the number of elements in the relation.

Smallest Equivalence Relation

The smallest equivalence relation is the identity relation, where each element is only related to itself. This is because:

  • It is reflexive (each element is related to itself),
  • It is trivially symmetric and transitive (since there are no other pairs besides (a,a)(a, a)).

For a set SS with nn elements, the number of ordered pairs in the smallest equivalence relation is simply the set of pairs where each element is related to itself. Hence, the number of ordered pairs in the smallest equivalence relation is nn (one pair for each element: (a1,a1),(a2,a2),,(an,an)(a_1, a_1), (a_2, a_2), \dots, (a_n, a_n)).

Largest Equivalence Relation

The largest equivalence relation is the universal relation where every element is related to every other element. In this case, the relation consists of all possible pairs of elements in SS. Since the relation must be reflexive, symmetric, and transitive, this means:

  • Every element is related to every other element in the set.

Thus, the number of ordered pairs in the largest equivalence relation is equal to the total number of pairs in S×SS \times S, which is n2n^2 (since there are nn elements, each element can be paired with nn elements, including itself).

Conclusion

  • The number of ordered pairs in the smallest equivalence relation is nn.
  • The number of ordered pairs in the largest equivalence relation is n2n^2.

Would you like more details on this explanation?

Here are 5 related questions:

  1. What are the conditions for a relation to be transitive?
  2. How do you determine if a relation is symmetric?
  3. Can there be more than one equivalence relation on a set with nn elements?
  4. How many equivalence classes are there in the largest equivalence relation?
  5. What is the number of ordered pairs in a partial equivalence relation?

Tip: An equivalence relation partitions a set into disjoint equivalence classes where every element in a class is related to every other element in that class.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Equivalence Relations
Cartesian Product

Formulas

Number of ordered pairs in the smallest equivalence relation = n
Number of ordered pairs in the largest equivalence relation = n^2

Theorems

Equivalence Relation Theorem

Suitable Grade Level

Undergraduate level